The General Theorem

In this section, we will show that in the special case where R is a polynomial ring in finitely many variables over a field, then the intersection algebra of two principal monomial ideals is a semigroup ring whose generators can be algorithmically computed.

Let k be a field and Q a semigroup. Recall that k[Q] is a semigroup ring, namely the

k-algebra withk-basis {ta_{|a∈Q}and multiplication defined byt}a_·tb _=ta+b_{. Also recall our}

notation: whenxis a homogeneous element in a semigroup ring, log(x) denotes its exponent vector, and ifX is a collection of homogeneous elements, log(X) refers to the set of exponent vectors of all the monomials in X.

Theorem 3.1.1. If R is a polynomial ring in n variables over k, and I and J are ideals generated by monomials (i.e. monic products of variables) in R, then B is a semigroup ring. Proof. Since I and J are monomial ideals, Ir_∩Js _{is as well for all r and s. So each (r, s)} component of B is generated by monomials, therefore B is a subring of k[x1, . . . , xn, u, v] generated over k by a list of monomials {bi|i ∈ Λ}. Let Q be the semigroup generated by

{log(bi)|i∈Λ}. ThenB=k[Q], andB is a semigroup ring overk.

Since a polynomial ring R = k[x1, . . . , xn] is a UFD, we will use the result of the pre-

vious chapter to show that intersection algebras of principal ideals in polynomial rings are finitely generated over R. Then, since R is generated over k by the variables x1, . . . , xn,

Theorem 3.1.2. Let I = (xa1

1 · · ·xann) and J = (x b1

1 · · ·xbnn) be principal ideals in R =

k[x1, . . . , xn], and let Σa,b be the fan associated to a= (a1, . . . , an) and b= (b1, . . . , bn). Let

Qi =Ci∩Z2 for every Ci ∈Σa,b

and HQi be its Hilbert basis of cardinality ni for all i = 0, . . . , n. Further, let Q be the subsemigroup in _N2 _{generated by}

{(a1rij, . . . , airij, bi+1sij, . . . , bnsij, rij, sij)|i= 0, . . . , n, j = 1, . . . , ni} ∪log(x1, . . . , xn),

where (rij, sij)∈ HQi for every i= 0, . . . n, j = 1, . . . , ni. Then B =k[Q]. Proof. Since R is a UFD, by Theorem 2.1.6, B is generated over R by

{xa1rij 1 · · ·x airij i x bi+1sij i+1 · · ·x bnsij n u rij_vsij_{|i= 0, . . . , n, j = 1, . . . , n} i}.

Then, since R is generated as an algebra over k by x1, . . . , xn, it follows that B ⊂

k[x1, . . . , xn, u, v] is generated as an algebra over k by the set

P ={x1, . . . , xn, x a1rij 1 · · ·x airij i x bi+1sij i+1 · · ·x bnsij n u rij_vsij_{|i= 0, . . . , n, j = 1, . . . , n} i}.

This is a set of monomials in k[x1, . . . , xn, u, v]. Now note that therefore

log(P) ={(a1rij, . . . , airij, bi+1sij, . . . , bnsij, rij, sij)|i= 0, . . . , n, j= 1, . . . , ni}

∪log(x1, . . . , xn).

In conclusion, log(P) =Q and hence B=k[Q].

Example 3.1.3. Let I = (x5_y2_{) and J = (x}2_y3_{). Then a}

and 5/2≥2/3. Then we have the following cones:

C0 ={λ1(0,1) +λ2(2,5)|λi ∈R≥0}

C1 ={λ1(2,5) +λ2(3,2)|λi ∈R≥0}

C2 ={λ1(3,2) +λ2(1,0)|λi ∈R≥0}

C0 is the wedge of the first quadrant between the y-axis and the vector (2,5), C1 is the

wedge between (2,5) and (3,2), and C3 is the wedge between (3,2) and (1,0). It is easy to

see that this fan fills the entire first quadrant. Intersecting these cones with _Z2 is equivalent

to only considering the integer lattice points in these cones.

The Hilbert Basis of Q0 = C0 ∩ Z2 is {(0,1),(1,3),(2,5)}, and their corresponding

monomials inB are given by the generators ofBr,s for each (r, s):

(0,1) : (I0∩J1)u= (x2y3)v −generator isx2y3v

(1,3) : (I1∩J3)uv3 = ((x5y2)∩(x6y9))uv3 = (x6y9)uv3−generator is x6y9uv3

(2,5) : (I2∩J5)u2v5 = ((x10y4)∩(x10y15))u2v5 = (x10y15)u2v5−generator is x10y15u2v5.

Notice that all the generator monomials are of the form xb1s_yb2s_ur_vs_{, with b}

1 = 2, b2 = 3,

and (r, s) is a Hilbert Basis element, as shown earlier.

The Hilbert Basis of Q1 is {(1,1),(1,2),(3,2),(2,5)}. In the same way as above, their

monomials arex5y3uv, x5y6uv2, x15y6u3v2, x10y15u2v5, all of which have the formxa1r_yb2s_ur_vs

with a1 = 5, b2 = 3 and (r, s) a basis element.

Lastly, the Hilbert Basis of Q2 is {(1,0),(2,1),(3,2)}, which gives rise to generators

x5y2u, x10y4u2, x15y6u3v2, all of which look like xa1r_ya2r_ur_vs _{with a}

1 = 5, a2 = 2.

Notice there are a few redundant generators in this list: those arise from lattice points that lie on the boundaries of the cones. So B is generated over R by

Then, since R is generated over k byx and y, B is generated over k by

{x, y, x5y2u, x10y4u2, x15y6u3v2, x5y3uv, x5y6uv2, x2y3v, x6y9uv3, x10y15u2v5}.

Using this technique, we have written a program in Macaulay2 that will provide the list of generators of B for any I and J. First it fan orders the exponent vectors, then finds the Hilbert Basis for each cone that arises from those vectors. Finally, it computes the corresponding monomial for each basis element. The code is below:

loadPackage "Polyhedra"

--function to get a list of exponent vectors from an ideal I expList=(I) ->(

flatten exponents first flatten entries gens I )

algGens=(I,J)->(

B:=(expList(J))_(positions(expList(J),i->i!=0)); A:=(expList(I))_(positions(expList(J),i->i!=0)); L:=sort apply(A,B,(i,j)->i/j);

C:=flatten {0,apply(L,i->numerator i),1}; D:=flatten {1, apply(L,i->denominator i),0}; M:=matrix{C,D};

G:=unique flatten apply (#C-1, i-> hilbertBasis (posHull submatrix(M,{i,i+1})));

S:=ring I[u,v];

flatten apply(#G,i->((first flatten entries gens

intersect(I^(G#i_(1,0)),J^(G#i_(0,0)))))*u^(G#i_(1,0))*v^(G#i_(0,0))) )